Mathematical Modeling-PCI

Level Process:

Consider the case of a simple level process described in Fig.2.

Let Q_i and Q_o are the inflow rate and outflow rate (in m³/sec) of the tank, and H is the height of the liquid level at any time instant. We assume that the cross sectional area of the tank be A. In a steady state, both Q_i and Q_o are same, and the height H of the tank will be constant. But when they are unequal, we can write,

Rate of accumulation in the tank= Inflow rate- Outflow rate

i.e., Q_i - Q_o = A dH/dT (1)

But the outflow rate Qo is dependent on the height of the tank.

Q_o = H/R where R= Resistance in flow (2)

We can also assume that the outlet pressure P2=0 (atmospheric pressure) and P1=Hgρ.

Thus, Qi – (H/R) = A (dH/dT) (3)

Or Qi = (H/R)+A (dH/dT) (4)

It can be easily seen, that equation (4) is a linear differential equation. So the transfer function of the process can easily be obtained as:

G(s) = H(s)/Q_i(s) = R/(ARs+1)= K_p/ ( t_ps+1)

where t_{p is} called the time constant and K_p the steady state gain.

It is to be noted that all the input and output variables in the transfer function model represent, the deviations from the steady state values. If the operating point (the steady state level H(s) in the present case) changes, the parameters of the process (K_p and t ) will also change.

It is to be noted that all the input and output variables in the transfer function model represent, the deviations from the steady state values. If the operating point (the steady state level H(s) in the present case) changes, the parameters of the process (K_p and t ) will also change.